Now if you view the JSON of the list of feeds the TeamCity feed should be at the top.

And that means the TeamCity plugin should push to it. It will probably display a message along the lines of "Skipping package push for repository". Just ignore that message; it should still work even though that message is shown.

[13:29:21]Skipping package push for repository http://teamcity:8111/httpAuth/app/nuget/v1/FeedService.svc/.[13:29:21]This respository is a TeamCity NuGet server which doesn't support pushing packages.[13:29:21]To publish the package to a TeamCity feed, run sqlCI as a TeamCity build step. This will publish the package as an artifact and add it to the TeamCity feed.

It's safe to ignore. If the build was successful, the build will be published to the TeamCity package feed.